Prices, Delay, and the Dynamics of Trade

We characterize the dynamics of trading patterns and market composition when trade is bilateral, finding a trading partner is costly, prices are determined by bargaining, and preferences are private information. We show that equilibrium is inefficient and exhibits delay as sellers price discriminate between buyers with different values. As frictions vanish, transaction prices are asymptotically competitive and the welfare loss of inefficient trading approaches zero, even though the trading patterns continue to be inefficient and delay persists. Journal of Economic Literature Classification Numbers: D40, D50.     

Ordinal Efficiency and the Polyhedral Separating Hyperplane Theorem

We study problems in which each of finitely many agents must be allocated a single object, based on the agents' rankings of pure outcomes. A random allocation is ordinally efficient if it is not ordinally dominated in the sense of there being another random assignment that gives each agent a first order stochastically dominant distribution of objects. We show that any ordinally efficient random assignment maximizes the sum of expected utilities for some vector of vNM utility functions that are consistent with the given ordinal preferences. One method of proof uses a new version of the separating hyperplane theorem for polyhedra. Journal of Economic Literature Classification Numbers: C78, D61.     

Legislative Bargaining and Coalition Formation

The finite horizon version of D. P. Baron and J. Ferejohn's [1989, Amer. Polit. Sci. Rev.83, 1181-1206] legislative bargaining model is investigated. With three or more periods, a continuum of divisions is supportable as subgame perfect equilib- ria. There exist equilibria where coalitions larger than a minimal winning coalition receive strictly positive shares. With sufficiently patient players and a sufficiently long horizon, any interior distribution is supportable as an equilibrium. In contrast, a generic uniqueness result applies when introducing heterogenous time preferences. The unique backwards induction equilibrium in the perturbed game is nonstationary, and neither the original (symmetric) nor the perturbed game provides guidance for equilibrium selection in the infinite game. Journal of Economic Literature Classification Numbers: C73, C78, D72, D78, H49.     

The role of independence in the Green-Lin Diamond-Dybvig model

Green and Lin study a version of the Diamond-Dybvig model with a finite number of agents, independence (independent determination of each agent's type), and sequential service. For special preferences, they show that the ex ante first-best allocation is the unique equilibrium outcome of the model with private information about types. Via a simple argument, it is shown that uniqueness of the truth-telling equilibrium holds for general preferences—and, in particular, for a constrained-efficient allocation whether first-best or not. The crucial assumption is independence.     

An Experimental Study of the Effect of Private Information in the Coase Theorem

We investigate, in an experimental setting, the effect of private information on the Coase theorem's predictions of efficiency and allocative neutrality. For a two-person bargaining game, we find significantly more inefficiency and allocative bias in the case of private information compared with the case of complete information. We also find substantial bargaining breakdown, which is not predicted by the Coase theorem. For the case of private information, we reject the Coase theorem in favor of the alternative of a generalized version of the Myerson—Satterthwaite theorem, which predicts inefficiency, allocative bias in the direction of the disagreement point, and some bargaining breakdown.     

Competitive bargaining equilibrium

In a simple exchange economy we propose a bargaining procedure that leads to a Walrasian outcome as the agents become increasingly patient. The competitive outcome therefore obtains even if agents have market power and are not price-takers. Moreover, where in other bargaining protocols the final outcome depends on bargaining power or relative impatience, the outcome here is determinate and depends only on preferences and endowments. Our bargaining procedure involves bargaining over prices and maximum quantity constraints, and it guarantees convergence to a Walrasian outcome for any standard exchange economy. In contrast, without quantity constraints we show that equilibrium is generically inefficient.     

Folk Theorem For Bilateral Bargaining with Vector Endowments

We analyse a bargaining game of two players on the division of the sum of their vector endowments, with alternating proposals and discounting of single period utilities. The pair of endowments is not weakly Pareto‐efficient. Until they reach an agreement, each of the parties can withdraw from bargaining and keep their endowments. Any strictly individually rational division of the sum of endowments can emerge in some subgame perfect equilibrium if discount factors are close enough to one. Each subgame perfect equilibrium, in which the parties’ decisions do not depend on past rejected proposals, leads to a weakly Pareto‐efficient agreement in the first period.     

A “Super” Folk Theorem for dynastic repeated games

We analyze dynastic repeated games. These are repeated games in which the stage game is played by successive generations of finitely-lived players with dynastic preferences. Each individual has preferences that replicate those of the infinitely-lived players of a standard discounted infinitely-repeated game. Individuals live one period and do not observe the history of play that takes place before their birth, but instead create social memory through private messages received from their immediate predecessors. Under mild conditions, when players are sufficiently patient, all feasible payoff vectors (including those below the minmax of the stage game) can be sustained by sequential equilibria of the dynastic repeated game with private communication. In particular, the result applies to any stage game with n  ≥  4 players for which the standard Folk Theorem yields a payoff set with a non-empty interior. We are also able to characterize fully the conditions under which a sequential equilibrium of the dynastic repeated game can yield a payoff vector not sustainable as a subgame perfect equilibrium of the standard repeated game. For this to be the case it must be that the players’ equilibrium beliefs violate a condition that we term “inter-generational agreement.” A previous version of this paper was circulated as Anderlini et al. (2005). We are grateful to Jeff Ely, Leonardo Felli, Navin Kartik, David Levine, Stephen Morris, Michele Piccione, Andrew Postlewaite, Lones Smith and to seminar audiences at Bocconi, Cambridge, CEPR-Guerzensee, Chicago, Columbia, Edinburgh, Essex, Georgetown, Leicester, LSE, Northwestern, Oxford, Rome (La Sapienza), Rutgers, SAET-Vigo, Stanford, SUNY-Albany, UCL, UC-San Diego, Venice and Yale for helpful feedback.     

Majority Rule in a Stochastic Model of Bargaining

In this paper we consider multilateral stochastic bargaining models with general agreement rules. For n-player games where in each period a player is randomly selected to allocate a stochastic level of surplus and q?n players have to agree on a proposal to induce its acceptance, we characterize the set of stationary subgame perfect equilibrium payoffs and establish their existence. We show that for agreement rules other than the unanimity rule, the equilibrium payoffs need not be unique. Furthermore, even when the equilibrium is unique, it need not be efficient. Journal of Economic Literature Classification Numbers: C73, C78, D70.     

Bargaining and Boldness

We study a multiperson bargaining problem with general risk preferences through the use of Shaked's game of cycling offers with exogenous breakdown. If preferences are “smooth,” then as the risk of breakdown vanishes, the limiting outcome is one in which bargainers are equally marginally bold; where a bargainer's marginal boldness measures his willingness to risk disagreement in return for a marginal improvement in his position. Under smoothness, any (ordinal-)Nash solution is an equally marginally bold outcome. However, unlike the concept of the (ordinal-)Nash solution, a unique equally marginally bold outcome exists in natural cases—in particular, if all bargainers have risk-averse preferences of the rank-dependent expected utility type. For these preferences, the equally marginally bold outcome maximizes a “bargaining power”-adjusted (asymmetric) Nash product where the degree of asymmetry is determined by the disparity in the marginal valuation of certainty among bargainers. Journal of Economic Literature Classification Numbers: C72, C78, D81.     

Can a larger market foster R&D under monopolistic competition with variable mark-ups?

We study monopolistic competition with symmetric directly additive preferences (generating variable mark-ups) and an endogenous technology choice. Each firm chooses an investment in R&D to decrease its marginal cost. We prove that the equilibrium R&D investment increases with market size (a larger population or trade) only if the price-elasticity of demand is an increasing function. Together with the output levels, such equilibrium investments may be socially excessive or insufficient, depending on whether the elasticity of the subutility is increasing or decreasing. The main implication is that opening up to free trade can foster R&D through variable mark-ups.     

Social Learning in a Common Interest Voting Game

This paper describes the experimental results of a dynamic voting game. Subjects are induced with common preferences, but have asymmetric and incomplete information about the payoff-generating behavior of two available proposals. The election process serves as a mechanism to gather the dispersed information. It is shown that subjects are more effective in coordinating their decisions to a socially optimal equilibrium when they get more experienced. The way they succeed in doing this is by paying less attention to the popularity of a proposal.Journal of Economic LiteratureClassification Numbers: D72, D82.     

Experts against adverse selection: A note on the existence of equilibrium with costly appropriable information

The existence of competitive equilibrium in Laffont's (J. Econ. Theory10 (1975)) model of adverse selection with costly information is studied. The existence of an equilibrium with finite prices is demonstrated without unusual restrictions on preferences or the technology of information production. This is made possible by changing the way in which the behaviour of information producing agents is modelled, and allowing for some public information.     

The Identification of Preferences from Equilibrium Prices under Uncertainty

The competitive equilibrium correspondence, which associates equilibrium prices of commodities and assets with allocations of endowments, identifies the preferences and beliefs of individuals under uncertainty; this is the case even if the asset market is incomplete. Journal of Economic Literature Classification Numbers: D52, D80.     

Bayesian Equilibrium in a Public Good Economy

This paper studies the provision of a public good via voluntary contributions in an economy with uncertainty and differential information. Consumers differ in their private information regarding their future endowment as well as in their preferences. Each consumer selects her consumption ex ante, i.e., before knowing the state of nature. Contributions to the provision of the public good are determined ex post, i.e., when the state of nature is realized. Assuming that some normality conditions hold, a Bayesian equilibrium exists. Further, equilibrium is unique, regardless of the number of consumers, when either (1) the information partitions of consumers can be ranked from the finest to the coarsest, or (2) there are only two types of consumers.     

Holdups and Efficiency with Search Frictions

A natural holdup problem arises in a market with search frictions: Firms have to make a range of investments before finding their employees, and larger investments translate into higher wages. In particular, when wages are determined by ex post bargaining, the equilibrium is always inefficient: Recognizing that capital-intensive production relations have to pay higher wages, firms reduce their investments. This can only be prevented by removing all the bargaining power from the workers, but this, in turn, depresses wages below their social product and creates excessive entry of firms. In contrast to this benchmark, we show that efficiency is achieved when firms post wages and workers can direct their search toward more attractive offers. This efficiency result generalizes to an environment with imperfect information where workers only observe a few of the equilibrium wage offers. We show that the underlying reason for efficiency is not wage posting per se, but the ability of workers to direct their search toward more capital-intensive jobs.     

Growth with many agents and wages paid ex ante

In this paper, we consider a one-sector model of economic growth with several infinitely-lived heterogeneous agents, who are endowed with diverse discount factors as well as preferences over consumption. In line with the classical Ramsey model, agents are not allowed to borrow against future income. Unlike the traditional assumption of ex post wage payment, wages are paid ex ante in our model. We first explain the difference between the assumptions of wages being paid ex ante and wages being paid ex post in the framework of a simple illustrative two-class model. Our main result shows that in contrast to the many-agent Ramsey model with ex post wage payment, the capital stock sequence converges to the steady state stock irrespective of production technology employed by the firms. Further, all impatient agents own zero capital stock, whereas the most patient agent owns the entire capital stock from some time onward. Thus, we have shown that a slight modification in the timing of wage payment in growth models can lead to significant changes in the stability properties of equilibrium dynamics.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

Passing the Buck On the acceptance of responsibility

In the Passing the Buck game, the n members of a group decide in a given order whether or not to incur costs so that they can fix a certain problem for the benefit of the group. Passing the Buck means that one relies on the service of a successor. The decision is influenced by social preferences and by strategic considerations. We derive three main conclusions from experiments with n=3 and n=4. (i) The unique perfect Baysian equilibrium of the Passing the Buck game with incomplete information about the preferences of other players generally fits the data, however with two interesting exceptions. (ii) Backward induction breaks down if it requires more than two steps, i.e., for the first player in the game with n=4. (iii) Making one of the members of the group an “expert” who alone is competent to fix the problem is connected with a change of social preferences.     

Implementation with partial provability

We extend implementation theory by allowing the social choice function to depend on more than just the preferences of the agents and allowing agents to support their statements with hard evidence. We show that a simple condition on evidence is necessary for the implementation of a social choice function f when the preferences of the agents are state independent and sufficient for implementation for any preferences (including state dependent) with at least three agents if the social planner can perform small monetary transfers beyond those called for by f. If transfers can be large, f can be implemented in a game with perfect information when there are at least two players under a boundedness assumption. For both results, transfers only occur out of equilibrium. The use of evidence enables implementation which is robust in the sense that the planner needs little information about agents? preferences or beliefs and agents need little information about each others? preferences. Our results are robust to evidence forgery at any strictly positive cost.